Word Count : 302

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Green’s theorem states that for a vector field \(\mathbf{F} = (P, Q)\), the line integral along a simple, closed contour \( C \) is given by: \[ \oint_C \left( P \,dx + Q \,dy \right) = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \] ### Given Integral: \[ \oint_C \left( y^3 \,dx + 3x^3 \,dy \right) \] Here, - \( P = y^3 \) - \( Q = 3x^3 \) ### __________ _________ ___ ________ __________ _____ __________.
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